Method and apparatus for pilot estimation using an adaptive prediction error method with a kalman filter and a gauss-newton algorithm

ABSTRACT

A system is disclosed for use in a wireless communication system to provide an estimated pilot signal. The system includes a receiver and a front-end processing and despreading component in electronic communication with the receiver for despreading a CDMA signal. A pilot estimation component is in electronic communication with the front-end processing and despreading component for estimating an original pilot signal using a time-varying adaptive pilot estimator that includes a Kalman filter to produce a pilot estimate. A demodulation component is in electronic communication with the pilot estimation component and the front-end processing and despreading component for providing demodulated data symbols. The Kalman filter is configured by an offline system identification process that calculates parameters using a prediction error method and a Gauss-Newton algorithm and generates state estimates using the Kalman filter. The calculating and generating are iteratively performed to train the Kalman filter for real-time operation.

RELATED APPLICATIONS Reference to Co-Pending Applications for Patent

[0001] The present invention is related to the following Applicationsfor Patent in the U.S. Patent & Trademark Office:

[0002] “Method And Apparatus For Pilot Estimation Using SuboptimumExpectation Maximization” by Farrokh Abrishamkar et al., having AttorneyDocket No. 020123, filed concurrently herewith and assigned to theassignee hereof;

[0003] “Method And Apparatus For Pilot Estimation Using A Wiener Filter”by Farrokh Abrishamkar et al., having Attorney Docket No. 020099, filedconcurrently herewith and assigned to the assignee hereof;

[0004] “Method And Apparatus For Pilot Estimation Using A PredictionError Method With A Kalman Filter And Pseudo-Linear Regression”, byFarrokh Abrishamkar et al., having Attorney Docket No. 020201, filedconcurrently herewith and assigned to the assignee hereof; and

[0005] “Method And Apparatus For Pilot Estimation Using A PredictionError Method With A Kalman Filter And A Gauss-Newton Algorithm”, byFarrokh Abrishamkar et al., having Attorney Docket No. 020205, filedconcurrently herewith and assigned to the assignee hereof.

FIELD

[0006] The present invention relates to wireless communication systemsgenerally and specifically, to methods and apparatus for estimating apilot signal in a code division multiple access system.

BACKGROUND

[0007] In a wireless radiotelephone communication system, many userscommunicate over a wireless channel. The use of code division multipleaccess (CDMA) modulation techniques is one of several techniques forfacilitating communications in which a large number of system users arepresent. Other multiple access communication system techniques, such astime division multiple access (TDMA) and frequency division multipleaccess (FDMA) are known in the art. However, the spread spectrummodulation technique of CDMA has significant advantages over thesemodulation techniques for multiple access communication systems.

[0008] The CDMA technique has many advantages. An exemplary CDMA systemis described in U.S. Pat. No. 4,901,307, entitled “Spread SpectrumMultiple Access Communication System Using Satellite Or TerrestrialRepeaters”, issued Feb. 13, 1990, assigned to the assignee of thepresent invention, and incorporated herein by reference. An exemplaryCDMA system is further described in U.S. Pat. No. 5,103,459, entitled“System And Method For Generating Signal Waveforms In A CDMA CellularTelephone System”, issued Apr. 7, 1992, assigned to the assignee of thepresent invention, and incorporated herein by reference.

[0009] In each of the above patents, the use of a forward-link (basestation to mobile station) pilot signal is disclosed. In a typical CDMAwireless communication system, such as that described in EIA/TIA IS-95,the pilot signal is a “beacon” transmitting a constant data value andspread with the same pseudonoise (PN) sequences used by the trafficbearing signals. The pilot signal is typically covered with the all-zeroWalsh sequence. During initial system acquisition, the mobile stationsearches through PN offsets to locate a base station's pilot signal.Once it has acquired the pilot signal, it can then derive a stable phaseand magnitude reference for coherent demodulation, such as thatdescribed in U.S. Pat. No. 5,764,687 entitled “Mobile DemodulatorArchitecture For A Spread Spectrum Multiple Access CommunicationSystem,” issued Jun. 9, 1998, assigned to the assignee of the presentinvention, and incorporated herein by reference.

[0010] Recently, third-generation (3G) wireless radiotelephonecommunication systems have been proposed in which a reverse-link (mobilestation to base station) pilot channel is used. For example, in thecurrently proposed cdma2000 standard, the mobile station transmits aReverse Link Pilot Channel (R-PICH) that the base station uses forinitial acquisition, time tracking, rake-receiver coherent referencerecovery, and power control measurements.

[0011] Pilot signals can be affected by noise, fading and other factors.As a result, a received pilot signal may be degraded and different thanthe originally transmitted pilot signal. Information contained in thepilot signal may be lost because of noise, fading and other factors.

[0012] There is a need, therefore, to process the pilot signal tocounter the effects of noise, fading and other signal-degrading factors.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013]FIG. 1 is a diagram of a spread spectrum communication system thatsupports a number of users.

[0014]FIG. 2 is a block diagram of a base station and a mobile stationin a communications system.

[0015]FIG. 3 is a block diagram illustrating the downlink and the uplinkbetween the base station and the mobile station.

[0016]FIG. 4 is a block diagram of the channels in an embodiment of thedownlink.

[0017]FIG. 5 illustrates a block diagram of certain components in anembodiment of a mobile station.

[0018]FIG. 6 is a flow diagram of one embodiment of a method forestimating the pilot using a Kalman filter.

[0019]FIG. 7 is a block diagram illustrating the use of an offlinesystem identification component to determine the parameters needed bythe Kalman filter.

[0020]FIG. 8 is a block diagram illustrating how state estimation may bebootstrapped with parameter estimation.

[0021]FIG. 9 is a block diagram illustrating the offline systemidentification operation.

[0022]FIG. 10 is a block diagram illustrating real-time operation of thepilot estimation component.

[0023]FIG. 11 is a flow diagram of a method for configuring a Kalmanfilter for steady state operation to estimate the pilot.

[0024]FIG. 12 is a block diagram illustrating the inputs to and outputsfrom the offline system identification component and pilot estimationcomponent.

[0025]FIG. 13 is a block diagram of pilot estimation where the filteringis broken down into its I and Q components.

DETAILED DESCRIPTION

[0026] The word “exemplary” is used exclusively herein to mean “servingas an example, instance, or illustration.” Any embodiment describedherein as “exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments. While the various aspects of theembodiments are presented in drawings, the drawings are not necessarilydrawn to scale unless specifically indicated.

[0027] The following discussion develops the exemplary embodiments of adata-driven time-varying adaptive pilot estimator by first discussing aspread-spectrum wireless communication system. Then components of anembodiment of a mobile station are shown in relation to providing apilot estimate. Before the pilot is estimated, a pilot estimationcomponent is trained. Details regarding the offline systemidentification used to train the pilot estimation component are setforth. Included in the specification relating to the offline systemidentification are illustrations and mathematical derivations for aprediction error method (PEM) based on an innovations representation(IR) model of the noisy faded pilot symbols. A Gauss-Newton-basedstochastic gradient learning algorithm is also discussed. The iterativeprocess of generating state estimates and calculating new parameters isdiscussed. Formulas for both offline system identification and real-timepilot estimating are illustrated.

[0028] Note that the exemplary embodiment is provided as an exemplarthroughout this discussion; however, alternate embodiments mayincorporate various aspects without departing from the scope of thepresent invention.

[0029] The exemplary embodiment employs a spread-spectrum wirelesscommunication system. Wireless communication systems are widely deployedto provide various types of communication such as voice, data, and soon. These systems may be based on CDMA, TDMA, or some other modulationtechniques. A CDMA system provides certain advantages over other typesof systems, including increased system capacity.

[0030] A system may be designed to support one or more standards such asthe “TIA/EIA/IS-95-B Mobile Station-Base Station Compatibility Standardfor Dual-Mode Wideband Spread Spectrum Cellular System” referred toherein as the IS-95 standard, the standard offered by a consortium named“3rd Generation Partnership Project” referred to herein as 3GPP, andembodied in a set of documents including Document Nos. 3G TS 25.211, 3GTS 25.212, 3G TS 25.213, and 3G TS 25.214, 3G TS 25.302, referred toherein as the W-CDMA standard, the standard offered by a consortiumnamed “3rd Generation Partnership Project 2” referred to herein as3GPP2, and TR-45.5 referred to herein as the cdma2000 standard, formerlycalled IS-2000 MC. The standards cited hereinabove are hereby expresslyincorporated herein by reference.

[0031] Each standard specifically defines the processing of data fortransmission from base station to mobile, and vice versa. As anexemplary embodiment the following discussion considers aspread-spectrum communication system consistent with the CDMA2000standard of protocols. Alternate embodiments may incorporate anotherstandard.

[0032]FIG. 1 serves as an example of a communications system 100 thatsupports a number of users and is capable of implementing at least someaspects of the embodiments discussed herein. Any of a variety ofalgorithms and methods may be used to schedule transmissions in system100. System 100 provides communication for a number of cells 102A-102G,each of which is serviced by a corresponding base station 104A-104G,respectively. In the exemplary embodiment, some of the base stations 104have multiple receive antennas and others have only one receive antenna.Similarly, some of the base stations 104 have multiple transmitantennas, and others have single transmit antennas. There are norestrictions on the combinations of transmit antennas and receiveantennas. Therefore, it is possible for a base station 104 to havemultiple transmit antennas and a single receive antenna, or to havemultiple receive antennas and a single transmit antenna, or to have bothsingle or multiple transmit and receive antennas.

[0033] Terminals 106 in the coverage area may be fixed (i.e.,stationary) or mobile. As shown in FIG. 1, various terminals 106 aredispersed throughout the system. Each terminal 106 communicates with atleast one and possibly more base stations 104 on the downlink and uplinkat any given moment depending on, for example, whether soft handoff isemployed or whether the terminal is designed and operated to(concurrently or sequentially) receive multiple transmissions frommultiple base stations. Soft handoff in CDMA communications systems iswell known in the art and is described in detail in U.S. Pat. No.5,101,501, entitled “Method and System for Providing a Soft Handoff in aCDMA Cellular Telephone System”, which is assigned to the assignee ofthe present invention.

[0034] The downlink refers to transmission from the base station 104 tothe terminal 106, and the uplink refers to transmission from theterminal 106 to the base station 104. In the exemplary embodiment, someof terminals 106 have multiple receive antennas and others have only onereceive antenna. In FIG. 1, base station 104A transmits data toterminals 106A and 106J on the downlink, base station 104B transmitsdata to terminals 106B and 106J, base station 104C transmits data toterminal 106C, and so on.

[0035]FIG. 2 is a block diagram of the base station 202 and mobilestation 204 in a communications system. A base station 202 is inwireless communications with the mobile station 204. As mentioned above,the base station 202 transmits signals to mobile stations 204 thatreceive the signals. In addition, mobile stations 204 may also transmitsignals to the base station 202.

[0036]FIG. 3 is a block diagram of the base station 202 and mobilestation 204 illustrating the downlink 302 and the uplink 304. Thedownlink 302 refers to transmissions from the base station 202 to themobile station 204, and the uplink 304 refers to transmissions from themobile station 204 to the base station 202.

[0037]FIG. 4 is a block diagram of the channels in an embodiment of thedownlink 302. The downlink 302 includes the pilot channel 402, the syncchannel 404, the paging channel 406 and the traffic channel 408. Thedownlink 302 illustrated is only one possible embodiment of a downlinkand it will be appreciated that other channels may be added or removedfrom the downlink 302.

[0038] Although not illustrated, the uplink 304 may also include a pilotchannel. Recall that third-generation (3G) wireless radiotelephonecommunication systems have been proposed in which an uplink 304 pilotchannel is used. For example, in the currently proposed cdma2000standard, the mobile station transmits a Reverse Link Pilot Channel(R-PICH) that the base station uses for initial acquisition, timetracking, rake-receiver coherent reference recovery, and power controlmeasurements. Thus, systems and methods herein may be used to estimate apilot signal whether on the downlink 302 or on the uplink 304.

[0039] Under one CDMA standard, described in the TelecommunicationsIndustry Association's TIA/EIA/IS-95-A Mobile Stations-Base StationCompatibility Standard for Dual-Mode Wideband Spread Spectrum CellularSystem, each base station 202 transmits pilot 402, sync 404, paging 406and forward traffic 408 channels to its users. The pilot channel 402 isan unmodulated, direct-sequence spread spectrum signal transmittedcontinuously by each base station 202. The pilot channel 402 allows eachuser to acquire the timing of the channels transmitted by the basestation 202, and provides a phase reference for coherent demodulation.The pilot channel 402 also provides a means for signal strengthcomparisons between base stations 202 to determine when to hand offbetween base stations 202 (such as when moving between cells).

[0040]FIG. 5 illustrates a block diagram of certain components in anembodiment of a mobile station 504. Other components that are typicallyincluded in the mobile station 504 may not be illustrated for thepurpose of focusing on the novel features of the embodiments herein.Many embodiments of mobile stations 504 are commercially available and,as a result, those skilled in the art will appreciate the componentsthat are not shown.

[0041] If the pilot channel 402 were being sent on the uplink 304, thecomponents illustrated may be used in a base station 202 to estimate thepilot channel. It is to be understood that the inventive principlesherein may be used with a variety of components to estimate a pilotwhether the pilot is being received by a mobile station 504, a basestation 202, or any other component in a wireless communications system.Thus, the embodiment of a mobile station 504 is an exemplary embodimentof the systems and methods but it is understood that the systems andmethods may be used in a variety of other contexts.

[0042] Referring again to FIG. 5, a spread spectrum signal is receivedat an antenna 506. The spread spectrum signal is provided by the antenna506 to a receiver 508. The receiver 508 down-converts the signal andprovides it to the front-end processing and despreading component 510.The front-end processing and despreading component 510 provides thereceived pilot signal 512 to the pilot estimation component 514. Thereceived pilot signal 512 typically includes noise and usually suffersfrom fading.

[0043] The front-end processing and despreading component 510 alsoprovides the traffic channel 516 to a demodulation component 518 thatdemodulates the data symbols.

[0044] The pilot estimation component 514 provides an estimated pilotsignal 520 to the demodulation component 518. The pilot estimationcomponent 514 may also provide the estimated pilot signal 520 to othersubsystems 522.

[0045] It will be appreciated by those skilled in the art thatadditional processing takes place at the mobile station 504. Theembodiment of the pilot estimation component 514 will be more fullydiscussed below. Generally, the pilot estimation component 514 operatesto estimate the pilot signal and effectively clean-up the pilot signalby reducing the noise and estimating the original pilot signal that wastransmitted.

[0046] Systems and methods disclosed herein use a Kalman filter toestimate the pilot signal. Kalman filters are known by those skilled inthe art. In short, a Kalman filter is an optimal recursive dataprocessing algorithm. A Kalman filter takes as inputs data relevant tothe system and estimates the current value(s) of variables of interest.A number of resources are currently available that explain in detail theuse of Kalman filters. A few of these resources are “Fundamentals ofKalman Filtering: A Practical Approach” by Paul Zarchan and HowardMusoff, “Kalman Filtering and Neural Networks” by Simon Haykin and“Estimation and Tracking: Principles, Techniques And Software” by YaakovBar-Shalom and X. Rong Li, all of which are incorporated herein byreference.

[0047]FIG. 6 is a flow diagram 600 of one embodiment of a method forestimating the pilot using a Kalman filter. The system receives 602 thebaseband CDMA signal. Then the front-end processing and despreadingcomponent 510 performs initial processing and despreading 604. Thereceived pilot signal is then provided 606 to the pilot estimationcomponent 514. The received pilot signal has been degraded by variouseffects, including noise and fading. The pilot estimation component 514estimates 608 the pilot channel using a Kalman filter. After the pilothas been estimated 608, it is provided 610 to the demodulation component518 as well as other subsystems 522.

[0048] Referring now to FIG. 7, before the Kalman filter in the pilotestimation component 514 is used, the parameters of the Kalman filterare determined during a training period. As shown, an offline systemidentification component 702 is used to determine the parameters neededby the Kalman filter. Offline training data is input to the offlinesystem identification component 702 in order to determine the neededparameters. Once the parameters have converged, they are provided to thepilot estimation component 714 and its Kalman filter, to process thereceived pilot and estimate the original pilot in real time. In theembodiment disclosed herein, the offline system identification component702 is used once to set up the parameters. After the parameters havebeen determined, the system uses the pilot estimation component 714 andno longer needs the offline system identification component 702.

[0049] Typically the offline system identification 702 is used before acomponent is being used by the end user. For example, if the system andmethods were being used in a mobile station 204, when an end user wasusing the mobile station 204, it 204 would be using the pilot estimationcomponent 714 to process the pilot in real-time. The offline systemidentification component 702 is used before the mobile station 204 isoperating in real-time to determine the parameters needed to estimatethe pilot.

[0050] The following discussion provides details regarding thecalculations that will be made in the offline system identificationcomponent 702 as well as the pilot estimation component 714. Additionaldetails and derivations known by those skilled in the art are notincluded herein.

[0051] The received pilot complex envelope after despreading is given bythe following formula:

{tilde over (y)} _(k) ={tilde over (s)} _(k) +{tilde over (v)}_(k)  Formula 1.

[0052] The received complex envelope in Formula 1 is represented as{tilde over (y)}_(k). The original but faded pilot signal is representedas {tilde over (s)}_(k). The noise component is represented as {tildeover (v)}_(k). For a single path mobile communication channel, theoriginal pilot signal may be represented by the mathematical model foundin Formula 2. The corresponding noise component may be represented bythe formula found in Formula 3.

{tilde over (s)} _(k)=ρ_(k) e ^(Jφ) ^(_(k)) R _(hh)(τ)=g _(k) N{squareroot}{square root over (E_(c) ^(p))} R _(hh)(τ){tilde over(f)}_(k)  Formula 2. $\begin{matrix}{{\overset{\sim}{}}_{k} = {{g_{k}\sqrt{{NI}_{oc}}{\overset{\sim}{n}}_{k}} + {g_{k}\sqrt{{NI}_{or}}{\sum\limits_{{m = {- \infty}},{m \neq k}}^{+ \infty}\quad {{R_{hh}\left( {{mT}_{c} - \tau} \right)}{{\overset{\sim}{w}}_{k}.}}}}}} & {{Formula}\quad 3}\end{matrix}$

[0053] The variables and parameters in the formulas found in Formulas 2and 3 are given in Table 1. TABLE 1 {square root}E^(p): Pilot EnvelopeI_(oc): Total AWGN Noise I_(or): Total Transmit PSD g_(k): AGC ControlSignal ρ_(k): Rice (Rayleigh) Fade Process {tilde over (f)}_(k): ComplexGaussian Fade Process with Clark Spectrum φ_(k): Fading Phase m, k: Chipand Symbol Counts N: Processing Gain R_(hh) (τ): Correlation τ: TimeOffset ñ_(k), {tilde over (w)}_(k): Zero Mean Unit Power Gaussian Noise

[0054] The demodulation component 518 requires the phase of the pilotsignal. In order to obtain the phase, the signals may be written in aform comprising I and Q components rather than being written in anenvelope form. In Formula 4, {tilde over (y)} represents the receivedpilot comprising its I and Q components. The faded pilot, without anynoise, is represented as {tilde over (s)} in Formula 5. The total noiseis represented in Formula 6 as {tilde over (v)}. Formula 7 illustratesthe fade as {tilde over (f)}.

{tilde over (y)}=y _(I) +jy _(Q)  Formula 4.

{tilde over (s)}=s _(I) +js _(Q)  Formula 5.

{tilde over (v)}=v _(I) +jv _(Q)  Formula 6.

{tilde over (f)}=ρe ^(jφ) =f _(I) +jf _(Q)  Formula 7.

[0055] Given the relationships of the formulas above, the I and Qcomponents of the faded pilot symbol without noise may be written asshown in Formulas 8 and 9.

s _(I)(k)=f _(I)(k)N{square root}{square root over (E_(c) ^(p))} R_(hh)(τ)g(k)  Formula 8.

s _(Q)(k)=f _(Q)(k)N{square root}{square root over (E_(c) ^(p))} R_(hh)(τ)g(k)  Formula 9.

[0056] Those skilled in the art will appreciate that the Wolddecomposition theorem may be used to model a time series. According toWold's decomposition, a time series can be decomposed into predictableand unpredictable components. The unpredictable component of the timeseries (under well-known spectral decomposition conditions) can beexpanded in terms of its innovations. The Wold expansion of observationsy_(k) may be approximated by a finite-dimensional ARMA Model as shown inFormula 10. The approximate innovations are represented by e_(k) and itis assumed that${E\left( e_{k} \middle| {\underset{\_}{Y}}_{k - 1} \right)} = 0.$

[0057] The optimal estimator may be propagated on Formula 10 resultingin three alternative forms as shown in Formulas 11, 13 and 16. Theapproximate innovations, represented by e_(k), is also the predictionerror, as shown in Formula 12. The equalities found in Formulas 14 and15 are assumed for Formula 13. Formulas 11, 13 and 16 are threealternative forms for the one-step predictor.

−y _(k) −a ₁ y _(k−1) − . . . −a _(n) y _(k−n) =e _(k) −d ₁ e _(k−1) − .. . −d _(m) e _(k−m)  Formula 10.

−ŷ _(k|k−1) =E(y _(k) |Y _(k−1))=a ₁ y _(k−1) + . . . +a _(n) y _(k−n)−d ₁ e _(k−1) − . . . −d _(m) e _(k−m)  Formula 11.

e _(k) =y _(k) −ŷ _(k|k−1)  Formula 12.

−ŷ _(k) =a ₁ ŷ _(k−1) + . . . +a _(n) ŷ _(k−n) +L ₁ e _(k−1) + . . . +L_(m) e _(k−m)  Formula 13.

ŷ _(k) =ŷ _(k|k−1)  Formula 14.

L _(i) =a ₁ −d ₁  Formula 15.

−ŷ _(k) =d ₁ ŷ _(k−1) + . . . +d _(n) ŷ _(k−n) +L ₁ y _(k−1) + . . . +L_(m) y _(k−m)  Formula 16.

[0058] Formulas 17-19 illustrate the first order (let x=y) one-steppredictors. Formula 17 is the first order one-step predictor thatcorresponds with Formula 11. Formula 18 is the first order one-steppredictor that corresponds with Formula 13. Formula 19 is the firstorder one-step predictor that corresponds with Formula 16.

−{circumflex over (x)} _(k+1) =ay _(k) −de _(k) =[y _(k) −e _(k) ][ad]^(T)  Formula 17.

−{circumflex over (x)} _(k+1) =a{circumflex over (x)} _(k) +Le _(k)=[{circumflex over (x)} _(k) e _(k) ][aL] ^(T)  Formula 18.

−{circumflex over (x)} _(k+1) =dŷ _(k) +Ly _(k) =[ŷ _(k) y _(k) ][dL]^(T)  Formula 19.

[0059] The alternative forms shown in Formulas 11, 13 and 16 are ARMAequivalents of a one-step Kalman Filter which may be seen in the firstorder case where m=n=1, a=d and L=L₁ yielding the equalities as shown inFormula 20. Since {circumflex over (x)}_(k)=ŷ_(k) indicates a firstorder Kalman filter state, a Kalman filter is provided as shown inFormula 21.

ŷ _(k+1) =aŷ _(k) +Le _(k)=(a−L)ŷ _(k) +Ly _(k) =dŷ _(k) +Ly_(k)  Formula 20.

{circumflex over (x)} _(k+1) =d{circumflex over (x)} _(k) +Ly_(k)  Formula 21.

[0060] In this embodiment, a prediction error method (PEM) is used.Unlike LMS or MMSE-based estimators, PEM does not need a preamble. PEMis completely data driven and can be used in blind communicationreceivers. Prediction error method involves finding optimum modelparameters a₁, d₁ and L₁ by minimizing a function of the one-stepprediction error, shown in Formula 22, with g being some cost function.Using this approach avoids the need of having an error based on theactual pilot signal.

[0061] The state estimation may be bootstrapped with parameterestimation, as shown in FIG. 8. The functional block diagram of FIG. 8is used to minimize the prediction error of the model output. The signaly_(k) is input into the Kalman filter 802 that provides an ARMA stateestimation and produces ŷ_(k). The prediction error, represented by e,is calculated and provided to the parameter estimation block 804. Theform regressor component 806 takes ŷ_(k) and e and provides input to thefilter parameter update component 808. In addition, the parameterestimation block 804 outputs θ and provides it to the Kalman filter 802as well. The model being used in the parameter estimation block isy_(k)=Φ_(k)({circumflex over (θ)}_(k))^(T)θ.

[0062] A quadratic loss function may be used as shown by Formula 23.Applying the quadratic loss function as shown in Formula 23 to Formula16 yields Formula 24. Formula 25 is a representation of a first ordermodel.

g(e _(k))=g(y _(k) −ŷ _(k|k−1)(θ))  Formula 22.

g(e _(k))=e _(k) ²=(y _(k) −ŷ _(k))²  Formula 23.

ŷ _(k) =[ŷ _(k−1) , . . . , ŷ _(k−n) , y _(k−1) , . . . , y _(k−m) ][d ₁, . . . , d _(n) , L ₁ , . . . , L _(m)]^(T) Δθ_(k−1)(θ)θ  Formula 24.

ŷ _(k) =[ŷ _(k−1) y _(k−1) ][dL] ^(T)  Formula 25.

[0063] The function φ_(k−1) in Formula 24 is a model-dependent functionof θ resulting from ŷ_(k)=ŷ_(k)(θ). It may be noted thatg(e_(k)(θ))=(y_(k)−φ_(k−1)(θ)θ)² is a non-quadratic in θ due to thefunction φ_(k−1)(θ). As a result a closed-form solution does not exist.

[0064] Relating to the adaptive PEM cost function for time-varyingapplications, a fading memory parameter, λ, may be used. The values forthe fading memory parameter may be bounded as shown in Formula 26. Giventhe range for λ as shown by Formula 26, the cost function may beexpressed as shown in Formula 27. Formula 28 shows the expression forγ_(k) and its approximation for very large k.

[0065] A recursive form for online computation of the cost function isshown in Formula 29. For very large k the recursive form of the costfunction may be expressed as shown in Formula 30.

0<λ≦1  Formula 26. $\begin{matrix}{{\Lambda_{k}(\theta)}\underset{\_}{\underset{\_}{\Delta}}\frac{1}{\gamma_{k}}{\sum\limits_{l = 1}^{k}\quad {\lambda^{k - l}{{e_{l}^{2}(\theta)}.}}}} & {{Formula}\quad 27} \\{\gamma_{k} = {{\sum\limits_{l = 1}^{k}\lambda^{k - l}} = {\frac{1 - \lambda^{k}}{1 - \lambda} \approx {\frac{1}{1 - \lambda}.}}}} & \text{Formula~~28} \\{{{{\hat{\Lambda}}_{k + 1}(\theta)} = {{\frac{{\lambda\gamma}_{k}}{\gamma_{k + 1}}{{\hat{\Lambda}}_{k}(\theta)}} + {\frac{1}{\gamma_{k + 1}}{e_{k + 1}^{2}(\theta)}}}},{{{\hat{\Lambda}}_{0}(\theta)} = 0.}} & {{Formula}\quad 29}\end{matrix}$

 {circumflex over (Λ)}_(k+1)(θ)=λ{circumflex over (Λ)}_(k)(θ)+(1−λ)e_(k+1) ²(θ), {circumflex over (Λ)}₀(θ)=0, 0<λ<1  Formula 30.

[0066] A generalized gradient descent algorithm has the forms as shownin Formulas 31-34. The expression for Δ{circumflex over (θ)}_(k) fromFormula 34 is shown in Formula 35. The expression for ∇_(θ)Λ({circumflexover (θ)}) used is shown in Formula 36. ∇_(θ)Λ({circumflex over (θ)}) isthe gradient of the PEM cost Λ(θ) evaluated at the current estimate{circumflex over (θ)}. The parameter α is the adjustable step size. Q isa symmetric positive definite matrix. Depending upon the value of Q, theresult may be a standard gradient descent (LMS-like) method or it may bea Newton method. When Q=1, the result is the standard gradient descent(LMS-like) method. When${Q = {\left( \frac{\partial^{2}{\Lambda \left( \hat{\theta} \right)}}{\partial\theta^{2}} \right)^{- 1} = ({Hessian})^{- 1}}},$

[0067] the result is the Newton method.

{circumflex over (θ)}_(k+1)={circumflex over (θ)}_(k) −αQ _(k)(

_(k))∇_(θ)Λ({circumflex over (θ)})  Formula 31. $\begin{matrix}{= {\hat{\theta} + {2\quad \alpha \quad {Q_{k}\left( {\hat{\theta}}_{k} \right)}\frac{1}{\gamma}{\sum\limits_{n = 1}^{k}{{\psi_{n - 1}^{T}\left( \hat{\theta} \right)}{e_{n}\left( {\hat{\theta}}_{n} \right)}{\lambda^{k - n}.}}}}}} & {{Formula}\quad 32}\end{matrix}$

 {circumflex over (θ)}_(k+1)={circumflex over (θ)}_(k) −αQ({circumflexover (θ)}_(k))∇_(θ)Λ({circumflex over (θ)}_(k))  Formula 33.

={circumflex over (θ)}_(k)+αΔ{circumflex over (θ)}_(k)  Formula 34.$\begin{matrix}{{\Delta {\hat{\theta}}_{k}} = {2{Q_{k}\left( {\hat{\theta}}_{k} \right)}\frac{1}{\gamma_{k}}{\sum\limits_{n = 1}^{k}{\psi_{k - 1}^{T}e_{k}{\lambda^{k - n}.}}}}} & {{Formula}\quad 35} \\{{\nabla_{\theta}{\Lambda \left( \hat{\theta} \right)}} = {2\left( {1 - \lambda} \right){\sum\limits_{n = 1}^{k}{\psi_{n - 1}^{T}{e_{n}\left( {\hat{\theta}}_{n} \right)}{\lambda^{k - n}.}}}}} & \text{Formula~~36}\end{matrix}$

[0068] The derivation of the approximation of the Hessian is shown inFormulas 37-40. The Gaussian-Newton approximation is shown in Formula40. The second term in Formula 39$\left( {{- \frac{2}{N}}{\sum\limits_{k = 1}^{k}{{e_{k}\left( \hat{\theta} \right)}\frac{\partial\quad}{\partial\theta}{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}}}} \right)$

[0069] is dropped to arrive at the approximation in Formula 40.$\begin{matrix}{{H\left( \hat{\theta} \right)} = {{\frac{\partial^{2}\quad}{\partial\theta^{2}}\Lambda \left( \hat{\theta} \right)} = {\frac{\partial\quad}{\partial\theta}{{\nabla_{\theta}{\Lambda \left( \hat{\theta} \right)}}.}}}} & {{Formula}\quad 37} \\{\quad {= {{\frac{2}{N}{\sum\limits_{k = 1}^{N}{{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}{\psi_{k - 1}\left( \hat{\theta} \right)}}}} - {\frac{2}{N}{\sum\limits_{k = 1}^{N}{\left( {y_{k} - {{{\hat{y}}_{k}\left( \hat{\theta} \right)}{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}\frac{\partial\quad}{\partial\theta}{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}}} \right).}}}}}} & {\text{Formula}\quad 38} \\{\quad {= {{\frac{2}{N}{\sum\limits_{k = 1}^{N}{{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}{\psi_{k - 1}\left( \hat{\theta} \right)}}}} - {\frac{2}{N}{\sum\limits_{k = 1}^{N}{{e_{k}\left( \hat{\theta} \right)}\frac{\partial\quad}{\partial\theta}{{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}.}}}}}}} & {\text{Formula}\quad 39} \\{\quad {\approx {\frac{2}{N}{\sum\limits_{k = 1}^{N}{{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}{{\psi_{k - 1}\left( \hat{\theta} \right)}.}}}}}} & {\text{Formula}\quad 40}\end{matrix}$

[0070] The Gauss-Newton method is obtained by approximating the Hessianas shown in Formula 41. In this case, Q is equal to that described byFormula 42. $\begin{matrix}{\frac{\partial^{2}{\Lambda \left( \hat{\theta} \right)}}{\partial\theta^{2}} \approx {\frac{2}{\gamma_{k}}{\sum\limits_{n = 1}^{k}{{\psi_{n - 1}^{T}\left( \hat{\theta} \right)}{\psi_{n - 1}\left( {\hat{\theta}}_{n} \right)}\lambda^{k - n}\underset{\_}{\underset{\_}{\Delta}}{\prod_{k}{\left( {\hat{\theta}}_{k} \right).}}}}}} & {{Formula}\quad 41}\end{matrix}$

 Q=Q({circumflex over (θ)}_(k))=Π_(k)({circumflex over(θ)}_(k))⁻¹  Formula 42.

[0071] The following formulas illustrate the gradient derivation.Formula 43 details ∇_(θ)Λ({circumflex over (θ)}). ψ is a row vector andis shown in Formula 44. $\begin{matrix}\begin{matrix}{{\nabla_{\theta}{\Lambda \left( \hat{\theta} \right)}} = \left( {\frac{\partial\quad}{\partial\theta}{\Lambda (\theta)}} \right)_{\theta = \hat{\theta}}^{T}} \\{= {{- \frac{2}{N}}{\sum\limits_{k = 1}^{N}{{\nabla_{\theta}{{\hat{y}}_{k}\left( \hat{\theta} \right)}}{e_{k}\left( \hat{\theta} \right)}}}}} \\{= {{- \frac{2}{N}}{\sum\limits_{k = 1}^{N}{{\nabla_{\theta}{\hat{y}}_{k}}\left( \hat{\theta} \right)\left( {y_{k} - {{\hat{y}}_{k}\left( \hat{\theta} \right)}} \right)}}}} \\{= {{- \frac{2}{N}}{\sum\limits_{k = 1}^{N}{{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}{e_{k}\left( \hat{\theta} \right)}}}}} \\{= {{- \frac{2}{N}}{\sum\limits_{k = 1}^{N}{{\psi_{k - 1}^{T}\left( \hat{\theta} \right)}{\left( {y_{k} - {{\hat{y}}_{k}\left( \hat{\theta} \right)}} \right).}}}}}\end{matrix} & {{Formula}\quad 43} \\{{\psi_{k - 1}(\theta)} = {{{- \frac{\partial\quad}{\partial\theta}}{e_{k}(\theta)}}\quad = {{\frac{\partial\quad}{\partial\theta}{{\hat{y}}_{k}(\theta)}}\quad = {\frac{\partial\quad}{\partial\theta}{\left( {{\varphi_{k - 1}(\theta)}\theta} \right).}}}}} & \text{Formula~~~44}\end{matrix}$

[0072] The required derivative of ŷ_(k)(θ)=φ_(k−1)(θ)θ with respect to θis given by Formulas 45 and 46. $\begin{matrix}{{{\psi_{k - 1}(\theta)}\underset{\_}{\underset{\_}{\Delta}}\frac{\partial\quad}{\partial\theta}{{\hat{y}}_{k{{k - 1}}}(\theta)}} = {\left( {{\frac{\partial\quad}{\partial\theta_{1}}{{\hat{y}}_{k{{k - 1}}}(\theta)}},\ldots \quad,{\frac{\partial\quad}{\partial\theta_{N}}{{\hat{y}}_{k{{k - 1}}}(\theta)}}} \right).}} & \text{Formula~~45} \\{{\psi_{k - 1}(\theta)} = {{\frac{\partial\quad}{\partial\theta}\left\lbrack {{\varphi_{k - 1}(\theta)}\theta} \right\rbrack} = {{\theta^{T}\left( {\frac{\partial\quad}{\partial\theta}{\varphi_{k - 1}^{T}(\theta)}} \right)} + {{\varphi_{k - 1}(\theta)}.}}}} & \text{Formula~~46}\end{matrix}$

[0073] In the first-order case, Formula 46 becomes the expression shownin Formula 47 where d=d₁. ψ_(k−2)(θ) and φ_(k−1)(θ) are row vectors. Itmay also be noted that ∇_(θ) is a column vector,$\frac{\partial\quad}{\partial\theta}$

[0074] is a row vector, and that$\nabla_{\theta}{= {\left( \frac{\partial\quad}{\partial\theta} \right)^{T}.}}$

 ψ_(k−1)(θ)=dψ _(k−2)(θ)+φ_(k−1)(θ)  Formula 47.

[0075] A parameter Π may be defined as shown in Formula 48. Theexpression as shown in Formula 48 is an unfaded form of Π_(k). Given therelationship as shown in Formula 48, the parameter Π_(k+1) may becalculated as shown in Formula 49. The expression for Π_(k+1) may bemodified to an Infinite Impulse Response (IIR) form to obtain thereal-time estimate of the Hessian, when 0<λ<1, as shown in Formula 50.The same λ is used as is used to modify the cost Λ_(k)(θ). ForGauss-Newton, Q=Π⁻¹. $\begin{matrix}{\prod_{k}{\underset{\_}{\underset{\_}{\Delta}}\frac{1}{k}{\sum\limits_{C = 1}^{k}\quad {\psi_{C - 1}^{T}\psi_{C - 1}\underset{\_}{\underset{\_}{\Delta}}\frac{1}{k}{\sum\limits_{n = 1}^{k}\quad {\psi_{n - 1}^{T}{\psi_{n - 1}.}}}}}}} & {{Formula}\quad 48} \\{\prod_{k + 1}{= {\frac{k}{k + 1}{\prod_{k}{{+ \frac{1}{k}}\psi_{k}^{T}{\psi_{k}.}}}}}} & \text{Formula~~49}\end{matrix}$

 Π_(k+1)=λΠ_(k)+(1−λ)ψ_(k) ^(T)ψ_(k)  Formula 50.

[0076] The parameter update is as shown in Formulas 51-53.∇_(θ)Λ({circumflex over (θ)}) may be written as shown in Formula 54.Assuming that {circumflex over (θ)}_(k)={circumflex over (θ)}_(k−1) . .. {circumflex over (θ)}₁≈θ*, then ∇_(θ)Λ(θ_(k−1)) is equal to that shownin Formula 55. Therefore, the parameter update becomes as shown inFormula 56. (The factor of 2 is absorbed into α.)

Δ

_(k) =αQ({circumflex over (θ)}_(k))∇_(θ)Λ({circumflex over(θ)}_(k))  Formula 51. $\begin{matrix}{= {\alpha \quad {Q\left( {\hat{\theta}}_{k} \right)}2\left( {1 - \lambda} \right){\sum\limits_{n = 1}^{k}\quad {\psi_{n - 1}^{T}e_{n}{\lambda^{k - n}.}}}}} & {{Formula}\quad 52}\end{matrix}$

 −Q=Π ⁻¹  Formula 53. $\begin{matrix}{{\nabla_{\theta}{\Lambda \left( {\hat{\theta}}_{k} \right)}} = {{2\left( {1 - \lambda} \right){\sum\limits_{n = 1}^{k - 1}\quad {\psi_{n - 1}^{T}e_{n}\lambda^{k - n}}}} + {2\left( {1 - \lambda} \right)\psi_{k - 1}{e_{k}.}}}} & {{Formula}\quad 54}\end{matrix}$

$\begin{matrix}{{\nabla_{\theta}{\Lambda \left( {\hat{\theta}}_{k - 1} \right)}} = {{2\left( {1 - \lambda} \right){\sum\limits_{n = 1}^{k - 1}\quad {\psi_{n - 1}^{T}e_{n}\lambda^{k - n}}}} \approx 0.}} & {{Formula}\quad 55}\end{matrix}$

 Δ{circumflex over (θ)}_(k) ≈αQ({circumflex over (θ)}_(k))(1−λ)ψ_(k−1) e_(k)  Formula 56.

[0077] The parameter update using a direct inversion of Π is shown inFormula 57. The Hessian estimate update is as shown in Formula 58.Formula 59 shows the filtered output. Formula 60 shows the predictionerror. The expression in Formula 61 is the parameter vector, and theexpression shown in Formula 62 is the regressor for filtering. Regardingthe innovation gradient update, the filtered regressor for adaptation isshown in Formulas 63-64.

{circumflex over (θ)}_(k)={circumflex over (θ)}_(k−1)+α_(k)(1−λ)Π_(k)⁻¹ψ_(k) ^(T) e _(k)  Formula 57.

Π_(k)=λΠ_(k)+(1−λ)ψ_(k) ^(T)ψ_(k)  Formula 58.

{circumflex over (x)} _(k) =ŷ _(k)=φ_(k) ^(T){circumflex over(θ)}_(k−1)  Formula 59.

e _(k) =y _(k) −{circumflex over (x)} _(k) =y _(k)−φ_(k) ^(T){circumflexover (θ)}_(k−1)  Formula 60.

θ_(k) =└â _(k) {circumflex over (L)} _(k)┘  Formula 61.

φ_(k) =[{circumflex over (x)} _(k−1) e _(k−1)]  Formula 62.

ψ_(k) ={circumflex over (d)} _(k−1)ψ_(k−1)+φ_(k)  Formula 63.

{circumflex over (d)} _(k) =â _(k) −{circumflex over (L)} _(k)  Formula64.

[0078] For the parameter update shown in Formula 65, R_(k) is equal tothat shown in Formula 66. It follows that R_(k) is equal to theexpression shown in Formula 67. Using the Matrix Inversion Lemma, theexpression in Formula 68 may be shown.

{circumflex over (θ)}_(k+1)={circumflex over (θ)}_(k)+α_(k) R_(k)ψ_(k−1) ^(T) e _(k)  Formula 65.

R _(k)=Π_(k) ⁻¹(1−λ)=(1−λ)[Π_(k−1)+(1−λ)ψ_(k−1) ^(T)ψ_(k−1)]⁻¹  Formula66. $\begin{matrix}{R_{k} = {\frac{1}{\lambda}{\left( {R_{k - 1} - \frac{R_{k - 1}\psi_{k - 1}^{T}\psi_{k - 1}R_{k - 1}}{\lambda + {\psi_{k - 1}R_{k - 1}\psi_{k - 1}^{T}}}} \right).}}} & {{Formula}\quad 67}\end{matrix}$

 (A+BCD)⁻¹ =A ⁻¹ −A ⁻¹ B(DA ⁻¹ B+C ⁻¹)⁻¹ DA ⁻¹  Formula 68.

[0079] A stability test may be performed prior to updating {circumflexover (θ)}_(k). One method for performing a stability test is to updatethe parameter {circumflex over (θ)}_(k) only when |a−L|<1.

[0080] To train the Kalman filter for real-time operation, an offlinefirst-order system identification loop is used. During this offlinefirst-order system identification loop, the data is represented as Y_(k)={y₁ . . . y_(k)}. This embodiment uses a first-order ARMA for ŷ_(k)and prediction e_(k)=y_(k)−ŷ_(k)=y_(k)−{circumflex over (x)}_(k). ARMAmodel-based estimators can require fewer computations than simple FIR(MA) or IIR (AR) estimators of equal performance. The one-step predictor(Kalman Filter) is obtained as shown in Formulas 69-73.

{circumflex over (x)} _(k) ⁺ ={circumflex over (d)}/{circumflex over(ax)} _(k) +{circumflex over (L)}/ây _(k)  Formula 69.

{circumflex over (x)} _(k+1) ={circumflex over (ax)} _(k) ⁺  Formula 70.

{circumflex over (φ)}_(k−1) =[{circumflex over (x)} _(k−1) y _(k−1) ]=[ŷ_(k−1) y _(k−1)]  Formula 71.

ψ_(k) ={circumflex over (d)}ψ _(k−1)+φ_(k−1) ,{circumflex over(d)}=â−{circumflex over (L)}  Formula 72. $\begin{matrix}{R_{k} = {\frac{1}{\lambda}{\left( {R_{k - 1} - \frac{R_{k - 1}\psi_{k - 1}^{T}\psi_{k - 1}R_{k - 1}}{\lambda + {\psi_{k - 1}R_{k - 1}\psi_{k - 1}^{T}}}} \right).}}} & {{Formula}\quad 73}\end{matrix}$

[0081] Minimizing the cost function results in the parameter update asrepresented in Formulas 74-76. The parameter {circumflex over (θ)}_(k+1)is equal to the expression shown in Formula 75 only when |{circumflexover (d)}_(k)|=|â_(k)−{circumflex over (L)}_(k)|<a.

Δ{circumflex over (θ)}_(k) =[ΔâΔ{circumflex over (L)}] ^(T)=(1−λ)R_(K)ψ_(k−1) ^(T) e _(k)  Formula 74.

{circumflex over (θ)}_(k+1)={circumflex over (θ)}_(k)+αΔ{circumflex over(θ)}_(k)  Formula 75.

−{circumflex over (θ)}_(k)

=[â

]  Formula 76.

[0082] For offline system identification of order one, the data is againrepresented as Y _(k)={y₁ . . . y_(k)}. A first-order ARMA is used fory_(k) and prediction e_(l)=y_(l)−ŷ_(l)=y_(l)−{circumflex over (x)}_(l).The one-step predictor (Kalman Filter) is obtained as shown in Formulas77-79 when ({circumflex over (L)}=â−{circumflex over (d)}) and l=1, . .. , k.

{circumflex over (x)} _(l+1|l) ={circumflex over (dx)} _(l|l+1)+{circumflex over (L)}y _(l)  Formula 77.

{circumflex over (φ)}_(l−1) =[{circumflex over (x)} _(l−1) y_(l−1)]  Formula 78.

ψ_(l) ={circumflex over (d)}ψ _(l−1)+φ_(l−1)  Formula 79.

[0083] The cost function may be minimized resulting in the parameterupdate according to Formulas 80-81. This update is subject to theadmissibility test. $\begin{matrix}{{\Delta \quad \hat{\theta}} = {\left\lbrack {{\Delta \quad \hat{d}},{\Delta \quad \hat{L}}} \right\rbrack^{T} = {\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}\quad {\psi_{l - 1}^{T}\psi_{l - 1}}}} \right)^{- 1}{\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}\quad {\psi_{l - 1}^{T}e_{l}}}} \right).}}}} & {{Formula}\quad 80}\end{matrix}$

 {circumflex over (θ)}_(k+1)={circumflex over (θ)}_(k)+αΔ{circumflexover (θ)}_(k)  Formula 81.

[0084] The pilot estimation component 714 operates to take as input thereceived pilot signal which is noisy and faded to produce an estimate ofthe pilot signal. A Kalman filter may be used in real-time to estimatethe pilot. In the training state, the Kalman filter is trained ontraining data. A parameter estimation component estimates parameters,discussed below, and provides the parameters to the Kalman filter. TheKalman filter uses the parameters and provides a state estimate to theparameter estimation component. The process shown is iterated throughuntil the parameters for the Kalman filter have converged. This processwill be more fully discussed in relation to FIGS. 9-13.

[0085]FIG. 9 is a block diagram illustrating the offline systemidentification operation 702. Initialized parameters are provided to theKalman filter 906 to generate state estimates. In addition, trainingdata (Y₁, Y₂, . . . Y_(N)) is also provided to the Kalman filter 906.With the initialized parameters and training data, the Kalman filter 906generates a state estimate according to the formulas as described above.The new state estimate is provided to the parameter estimation component910.

[0086] The parameter estimation component 910 calculates new parametervalues using the equation in Formula 80. A state space model is formed,and the Kalman filter 806 generates new sequence state estimate. Aparameter adjustment component 911 may adjust the parameter {circumflexover (θ)} according to the formula shown in the parameter adjustmentcomponent 811 of FIG. 9 if |{circumflex over (d)}|<1. The Kalman filter906 and the parameter estimation component 910 continue to operate untilthe parameters have converged.

[0087] In the embodiment of FIG. 9, the training runs for the length ofthe pilot symbol record. In addition, the sequence of pilot symbols maybe tuned to the target speed and environment of choice.

[0088]FIG. 10 is a block diagram illustrating the use of a first-orderARMA for ŷ_(k) and prediction e_(k)=y_(k)−ŷ_(k)=y_(k)−{circumflex over(x)}_(k). The parameters ψ_(k+1), e_(k+1), R_(k+1) are calculated asshown and provided to the parameter estimation component 1010.

[0089] The parameter estimation component 1010 calculates new parametervalues using the equation in Formula 74. A parameter adjustmentcomponent 1009 may adjust the parameter {circumflex over (θ)} accordingto the formula shown in the parameter adjustment component 1009 of FIG.10 subject to the parameter admissibility test. According to theparameter admissibility test, the parameter adjustment component 1009may adjust the parameter {circumflex over (θ)} if |{circumflex over(d)}|<1, otherwise {circumflex over (θ)} keeps its previous value.

[0090]FIG. 11 is a flow diagram of a method for configuring a Kalmanfilter for steady state operation to estimate the pilot. Trainingsamples are provided 1102 to the offline system identification component702. The parameters are initialized 1104. In addition, the state isinitialized 1106. Then the Kalman filter is used to generate 1108 a newstate estimate. The parameter estimation 1110 is used to generate 1110new parameters. The generating steps 1108, 1110 are repeated 1112 untilthe filter and parameters have converged. Those skilled in the art willappreciate the various ways in which one may determine that the filterand parameters have converged. After the system has completed trainingthe filter, the converged parameters are provided 1114 for onlinesteady-state (real-time) Kalman filter operation.

[0091]FIG. 12 is a block diagram illustrating the inputs to and outputsfrom the offline system identification component 702 and pilotestimation component 714. The offline system identification component702 is provided training samples Y _(N) and initial condition{circumflex over (x)}₀. The system identification component 702 operatesin an iterative fashion, as described above, until the necessaryparameters have converged. After the system identification component 702has completed training, it 702 provides the state, parameters andinitial conditions to the pilot estimation component 714. The pilotestimation component 714 comprises the Kalman filter operating inreal-time. The pilot estimation component 714, as described herein,includes a time-varying adaptive pilot estimator and estimates thepilot, given the received pilot as input.

[0092] As discussed above, the pilot estimation component 714 uses aKalman filter to estimate the pilot. The calculations for the Kalmanfilter operating in real-time and for the parameter estimations andadjustments are shown and discussed above. The Kalman filter is providedthe online received pilot symbols and estimates the pilot. As shown, theKalman filter produces an estimate for both the I and Q components ofthe pilot signal.

[0093]FIG. 13 is a block diagram of pilot estimation where the filteringis broken down into its I and Q components. The system identificationcomponent 702, using a Recursive Prediction Error Method, MaximumLikelihood and a Gauss-Newton algorithm (RPEM-ML-GN) as described above,provides the initial conditions to the pilot estimation component 802.As shown, the processing for the I component is similar to theprocessing for the Q component. The particular component is provided tothe pilot estimation component 802. The pilot estimation component 802generates an estimated pilot for that component. The pilot estimate isthen provided to the demodulation component 518 as well as othersubsystems 522.

[0094] Use of the present systems and methods to estimate the pilotsignal may be used for many different kinds of situations. One situationwhere the embodiments herein may be useful is when a user is moving athigh speeds. For example, if the user were aboard a bullet train his orher speed on the train may reach speeds of approximately 500 km/hr.Estimating the pilot signal using the disclosed embodiments in suchsituations may provide better results than other currently used methods.

[0095] Those of skill in the art would understand that information andsignals may be represented using any of a variety of differenttechnologies and techniques. For example, data, instructions, commands,information, signals, bits, symbols, and chips that may be referencedthroughout the above description may be represented by voltages,currents, electromagnetic waves, magnetic fields or particles, opticalfields or particles, or any combination thereof.

[0096] Those of skill would further appreciate that the variousillustrative logical blocks, modules, circuits, and algorithm stepsdescribed in connection with the embodiments disclosed herein may beimplemented as electronic hardware, computer software, or combinationsof both. To clearly illustrate this interchangeability of hardware andsoftware, various illustrative components, blocks, modules, circuits,and steps have been described above generally in terms of theirfunctionality. Whether such functionality is implemented as hardware orsoftware depends upon the particular application and design constraintsimposed on the overall system. Skilled artisans may implement thedescribed functionality in varying ways for each particular application,but such implementation decisions should not be interpreted as causing adeparture from the scope of the present invention.

[0097] The various illustrative logical blocks, modules, and circuitsdescribed in connection with the embodiments disclosed herein may beimplemented or performed with a general purpose processor, a digitalsignal processor (DSP), an application specific integrated circuit(ASIC), a field programmable gate array (FPGA) or other programmablelogic device, discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. A general purpose processor may be a microprocessor,but in the alternative, the processor may be any conventional processor,controller, microcontroller, or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

[0098] The steps of a method or algorithm described in connection withthe embodiments disclosed herein may be embodied directly in hardware,in a software module executed by a processor, or in a combination of thetwo. A software module may reside in RAM memory, flash memory, ROMmemory, EPROM memory, EEPROM memory, registers, hard disk, a removabledisk, a CD-ROM, or any other form of storage medium known in the art. Anexemplary storage medium is coupled to the processor such the processorcan read information from, and write information to, the storage medium.In the alternative, the storage medium may be integral to the processor.The processor and the storage medium may reside in an ASIC. The ASIC mayreside in a user terminal. In the alternative, the processor and thestorage medium may reside as discrete components in a user terminal.

[0099] The previous description of the disclosed embodiments is providedto enable any person skilled in the art to make or use the presentinvention. Various modifications to these embodiments will be readilyapparent to those skilled in the art, and the generic principles definedherein may be applied to other embodiments without departing from thespirit or scope of the invention. Thus, the present invention is notintended to be limited to the embodiments shown herein but is to beaccorded the widest scope consistent with the principles and novelfeatures disclosed herein.

What is claimed is:
 1. In a wireless communication system, a method forestimating an original pilot signal, the method comprising: receiving aCDMA signal; despreading the CDMA signal; obtaining a pilot signal fromthe CDMA signal; and estimating an original pilot signal using atime-varying adaptive pilot estimator that includes a Kalman filter toproduce a pilot estimate, wherein the Kalman filter is determinedthrough use of a Gauss-Newton algorithm.
 2. The method as in claim 1,wherein the CDMA signal is transmitted on a downlink and wherein thedownlink comprises a pilot channel.
 3. The method as in claim 1, whereinthe CDMA signal is transmitted on an uplink and wherein the uplinkcomprises a pilot channel.
 4. The method as in claim 1, furthercomprising demodulating the pilot estimate.
 5. The method as in claim 1,wherein the Kalman filter was configured by an offline systemidentification process.
 6. The method as in claim 5, wherein the offlinesystem identification process comprises: providing training samples; andcalculating parameters using a prediction error method and theGauss-Newton algorithm and generating a state estimate using the Kalmanfilter, wherein the calculating and generating are iteratively performeduntil the Kalman filter converges.
 7. The method as in claim 6, whereinthe parameters are calculated according to the following:${\Delta \quad \hat{\theta}} = {\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}\psi_{l - 1}}}} \right)^{- 1}\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}e_{l}}}} \right)}$


8. The method as in claim 7, wherein the prediction error method isbased on an innovations representation/model of the pilot signal.
 9. Themethod as in claim 7, wherein the prediction error method finds optimummodel parameters by minimizing a function of the one-step predictionerror.
 10. The method as in claim 9, wherein the Gauss-Newton algorithmis used in finding a numerical solution for the function.
 11. The methodas in claim 10, wherein the parameters are adjusted when |{circumflexover (d)}|<1 according to the following: {circumflex over(θ)}_(l+1)={circumflex over (θ)}_(l)+αΔ{circumflex over (θ)}_(l)
 12. Ina mobile station for use in a wireless communication system, a methodfor estimating an original pilot signal, the method comprising:receiving a CDMA signal; despreading the CDMA signal; obtaining a pilotsignal from the CDMA signal; and estimating an original pilot signalusing a time-varying adaptive pilot estimator that includes a Kalmanfilter to produce a pilot estimate, wherein the Kalman filter isdetermined through use of a Gauss-Newton algorithm.
 13. The method as inclaim 12, wherein the CDMA signal is transmitted on a downlink andwherein the downlink comprises a pilot channel.
 14. The method as inclaim 12, further comprising demodulating the pilot estimate.
 15. Themethod as in claim 12, wherein the Kalman filter was configured by anoffline system identification process.
 16. The method as in claim 15,wherein the offline system identification process comprises: providingtraining samples; and calculating parameters using a prediction errormethod and the Gauss-Newton algorithm and generating a state estimateusing the Kalman filter, wherein the calculating and generating areiteratively performed until the Kalman filter converges.
 17. The methodas in claim 16, wherein the parameters are calculated according to thefollowing:${\Delta \quad \hat{\theta}} = {\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}\psi_{l - 1}}}} \right)^{- 1}\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}e_{l}}}} \right)}$


18. The method as in claim 17, wherein the prediction error method isbased on an innovations representation model of the pilot signal. 19.The method as in claim 17, wherein the prediction error method findsoptimum model parameters by minimizing a function of the one-stepprediction error.
 20. The method as in claim 19, wherein theGauss-Newton algorithm is used in finding a numerical solution for thefunction.
 21. The method as in claim 20, wherein the parameters areadjusted when |{circumflex over (d)}|<1 according to the following:{circumflex over (θ)}_(l+1)={circumflex over (θ)}_(l)+αΔ{circumflex over(θ)}_(l)
 22. A mobile station for use in a wireless communication systemwherein the mobile station is configured to estimate an original pilotsignal, the mobile station comprising: an antenna for receiving a CDMAsignal; a receiver in electronic communication with the antenna; afront-end processing and despreading component in electroniccommunication with the receiver for despreading the CDMA signal; a pilotestimation component in electronic communication with the front-endprocessing and despreading component for estimating an original pilotsignal using a time-varying adaptive pilot estimator that includes aKalman filter to produce a pilot estimate, wherein the Kalman filter isdetermined through use of a Gauss-Newton algorithm; and a demodulationcomponent in electronic communication with the pilot estimationcomponent and the front-end processing and despreading component forproviding demodulated data symbols to the mobile station.
 23. The mobilestation as in claim 22, wherein the receiver receives the CDMA signaltransmitted on a downlink and wherein the downlink comprises a pilotchannel.
 24. The mobile station as in claim 22, wherein the Kalmanfilter was configured by an offline system identification process. 25.The mobile station as in claim 24, wherein the offline systemidentification process comprises: providing training samples; andcalculating parameters using a prediction error method and theGauss-Newton algorithm and generating a state estimate using the Kalmanfilter, wherein the calculating and generating are iteratively performeduntil the Kalman filter converges.
 26. The mobile station as in claim25, wherein the parameters are calculated according to the following:${\Delta \quad \hat{\theta}} = {\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}\psi_{l - 1}}}} \right)^{- 1}\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}e_{l}}}} \right)}$


27. The mobile station as in claim 26, wherein the prediction errormethod is based on an innovations representation model of the pilotsignal.
 28. The mobile station as in claim 26, wherein the predictionerror method finds optimum model parameters by minimizing a function ofthe one-step prediction error.
 29. The mobile station as in claim 28,wherein the Gauss-Newton algorithm is used in finding a numericalsolution for the function.
 30. The method as in claim 29, wherein theparameters are adjusted when |{circumflex over (d)}|<1 according to thefollowing: {circumflex over (θ)}_(l+1)={circumflex over(θ)}_(l)+αΔ{circumflex over (θ)}_(l)
 31. A mobile station for use in awireless communication system wherein the mobile station is configuredto estimate an original pilot signal, the mobile station comprising:means for receiving a CDMA signal; means for despreading the CDMAsignal; means for obtaining a pilot signal from the CDMA signal; andmeans for estimating an original pilot signal using a time-varyingadaptive pilot estimator that includes a Kalman filter to produce apilot estimate, wherein the Kalman filter is determined through use of aGauss-Newton algorithm.
 32. The mobile station as in claim 31, whereinthe CDMA signal is transmitted on a downlink and wherein the downlinkcomprises a pilot channel.
 33. The mobile station as in claim 31,further comprising means for demodulating the pilot estimate.
 34. Themobile station as in claim 31, wherein the Kalman filter was configuredby an offline system identification process.
 35. The mobile station asin claim 34, wherein the offline system identification processcomprises: providing training samples; and calculating parameters usinga prediction error method and the Gauss-Newton algorithm and generatinga state estimate using the Kalman filter, wherein the calculating andgenerating are iteratively performed until the Kalman filter converges.36. The mobile station as in claim 35, wherein the parameters arecalculated according to the following:${\Delta \quad \hat{\theta}} = {\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}\psi_{l - 1}}}} \right)^{- 1}\left( {\frac{1}{k}{\sum\limits_{l = 1}^{k}{\psi_{l - 1}^{T}e_{l}}}} \right)}$


37. The mobile station as in claim 36, wherein the prediction errormethod is based on an innovations representation model of the pilotsignal.
 38. The mobile station as in claim 37, wherein the predictionerror method finds optimum model parameters by minimizing a function ofthe one-step prediction error.
 39. The mobile station as in claim 38,wherein the Gauss-Newton algorithm is used in finding a numericalsolution for the function.
 40. The method as in claim 39, wherein theparameters are adjusted when |{circumflex over (d)}|<1 according to thefollowing: {circumflex over (θ)}_(l+1)={circumflex over(θ)}_(l)+αΔ{circumflex over (θ)}_(l)